The only conceivable thread that could knit those things together is the BCM. A/C Compressor requests are authorized by the BCM before being passed up to the ECM and then IPDM, locking on park or shutdown is directly controlled by the BCM, and an inaccurate battery current reading could cause the alternator to run harder than needed and compromise your gas mileage. BUT (capital but), that assumes they're all really related and that the BCM programming has basically gone cuckoo. The latter is very unlikely. So you should really get the car scanned with a good scanner that can detect all or most of the systems in the car, specifically at least the BCM, HVAC and IPDM in addition to the engine. Short of that, here are a few easy things to check:
1) The A/C compressor is a swash-plate type, so you'll never hear it turn on and off by itself like a cycling-clutch system. Basically, the only time you should hear the clutch cycle on or off is if you push the "AC" button on the HVAC display. If you do hear it when you toggle the button, the compressor isn't your issue. If you don't hear it, there are a lot of things that have to be right for the compressor to turn on, with 4 different devices involved. See this thread for some of them:
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2) The lock/unlock function for the doors is enabled and controlled from the menu in the cluster. There are three options, off (no locking), shift to park, and key-off. It's possible the stored values just got lost or corrupted and now your system is set to Off.
3) Especially if you find that 2 above is true, check your battery connections including all three connections on the ground cable, battery, chassis, transmission (block). Don't trust a visual inspection for the plus terminal, that style of lug can corrode on the inside with little or no evidence on the outside. See this thread for how to perform proper voltage-drop testing:
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